Final answer:
The question deals with the envelope of SSB and DSB-SC signals in the context of signal processing, as well as the intensity and properties of sinusoidal electromagnetic waves and wave functions in physics.
Step-by-step explanation:
The student's question pertains to the envelope of an SSB signal, the envelope of a DSB-SC signal, and various phenomena related to wave equations and intensities of sinusoidal electromagnetic waves. Starting with the Hilbert transform of a signal x(t), we note that the envelope of an SSB signal is given by the square root of the sum of the squares of the modulating signal and its Hilbert transform. That is, √x(t)² + ˆx(t)², where ˆx(t) is the Hilbert transform of x(t). For a DSB-SC signal with a cosine carrier, the envelope is the absolute value of the modulating signal, |x(t)|. Regarding the continuous sinusoidal electromagnetic wave, the peak intensity is twice the average intensity because the peak values of the electric and magnetic fields (Eo and Bo) are √2 times their root mean square (rms) values.
Further discussions also include the inverse square law for electromagnetic wave intensity, the inverse proportionality of wave fields to the distance from the source, superposition of sinusoidal waves, and the Born interpretation of the wave function's probability density. The vibrational energy in simple harmonic motion also gets addressed, utilizing the trigonometric identity that for any angle θ, cos²θ + sin²θ = 1, to obtain the total mechanical energy.